Introduction to the quantum Yang Baxter equation and quantum groups: an algebraic approach
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Dordrecht [u.a.]
Kluwer
1997
|
| Schriftenreihe: | Mathematics and its applications
423 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XX, 293 S. |
| ISBN: | 0792347218 |
Internformat
MARC
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| 100 | 1 | |a Lambe, Larry A. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Introduction to the quantum Yang Baxter equation and quantum groups |b an algebraic approach |c by Larry A. Lambe and David E. Radford |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer |c 1997 | |
| 300 | |a XX, 293 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematics and its applications |v 423 | |
| 650 | 4 | |a Mathematische Physik | |
| 650 | 4 | |a Hopf algebras | |
| 650 | 4 | |a Mathematical physics | |
| 650 | 4 | |a Quantum groups | |
| 650 | 4 | |a Yang-Baxter equation | |
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| 650 | 0 | 7 | |a Yang-Baxter-Gleichung |0 (DE-588)4291478-4 |2 gnd |9 rswk-swf |
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| 830 | 0 | |a Mathematics and its applications |v 423 |w (DE-604)BV008163334 |9 423 | |
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Datensatz im Suchindex
| _version_ | 1804126997782200320 |
|---|---|
| adam_text | Contents
Foreword xi
Preface xv
Acknowledgments xvii
Introduction xix
1. ALGEBRAIC PRELIMINARIES 1
1.1 Coalgebras 1
1.2 The Algebra C* 10
1.3 The Coalgebra A 17
1.3.1 The Construction and Characterizations o A0 17
1.3.2 Double Duals 20
1.3.3 The Fundamental Theorem of Coalgebras 21
1.4 Rational Modules and Comodules 23
1.4.1 Rational Modules 23
1.4.2 Comodules 24
1.4.3 MrandMr 27
1.4.4 MT Characterized in Terms of Annihilators 27
1.4.5 Another Proof of the Fundamental Theorem of Coalgebras 29
1.5 Bialgebras 32
1.6 Hopf Algebras 39
1.6.1 The Convolution Algebra 40
1.6.2 Definition of Hopf Algebra and Antipode 41
1.7 The Coradical and the Coradical Filtration 46
1.8 Pointed Hopf Algebras 53
1.9 (Co)Module (Co)Algebras 54
1.9.1 i? Module Algebras and Coalgebras 55
1.9.2 /f Comodule Algebras and Coalgebras 59
viii INTRODUCTION TO THE QYBE
2. THE QUANTUM YANG BAXTER EQUATION (QYBE) 65
2.1 The Constant Form of the QYBE 66
2.1.1 The Constant Form of the QYBE in H S Notation 67
2.1.2 The Constant Form of the QYBE in Coordinates 67
2.2 The Braid Equation 68
2.3 Symmetries 70
2.4 The One Parameter Form of the QYBE 72
2.5 The Two Parameter Form of the QYBE 74
2.6 A System of Polynomial Equations (the QYB Variety) 74
2.7 The Bialgebra Associated to the QYBE 76
2.7.1 A Module Action Associated to a QYBE Solution 76
2.7.2 Comodule Coaction 77
2.8 Factoring a QYBE Solution Over a Bialgebra 78
2.9 Compatibility Conditions in the Constant Case 79
2.9.1 The Fundamental Compatibility Condition in Coordinates 79
2.9.2 The (Co)Commutative Compatibility Condition 80
2.9.3 Compatibility Conditions in H S Notation 81
2.10 Compatibility Conditions in the Parameterized Cases 81
2.11 Reducing the Degree of the QYB Variety 83
2.11.1 From Cubic to Quadratic to Linear 83
2.11.2 A Curious Example 83
3. CATEGORIES OF QUANTUM YANG BAXTER MODULES 87
3.1 Various Categories 87
3.1.1 Left QYB A Modules 88
3.1.2 CQYB A Modules 89
3.1.3 Right QYB ^ Modules 90
3.1.4 Weak QYB/1 Modules 91
3.2 Congruence in A QyB 93
3.3 Recollections of Various Module and Comodule Structures 94
3.4 General Constructions in A QyB 96
3.4.1 Sub Objects, Quotient Objects of A QyB 96
3.4.2 Direct Sums in A QyB 96
3.4.3 Duals of Objects of AQyB 96
3.4.4 Structure Induced from Objects of A QyB 99
3.5 Constructions in nQyB when Hop has an Antipode 99
3.5.1 Equivalent Formulations of Compatibility 99
3.5.2 The Rational Part of a Left H, /P Module 101
3.5.3 Direct Products in AQyB 102
3.5.4 Sub Objects of Objects of H QyB when H°p has an Antipode 103
3.6 The Relationship Between QYBE Solutions R and RT 104
3.7 QYB Structures on H when H°p is a Hopf Algebra 105
CONTENTS ix
3.7.1 Generalized Coadjoint Action 106
3.7.2 Generalized Adjoint Action 109
3.8 Tensor Product in A QyB 110
3.8.1 The Tensor Algebra 113
3.8.2 Uom(M,N) and Quantum Yang Baxter Submodules 113
3.9 Tensor Product of Parameterized QYBE Solutions 114
3.10 Algebras of HQyB 115
3.11 Coalgebras, Bialgebras, and Hopf Algebras of nQy 116
3.12 Smash Biproducts Associated to H H QyB 117
4. MORE ON THE BIALGEBRA ASSOCIATED TO THE QYBE 121
4.1 Module Comodule Compatibility Revisited 121
4.2 A Basis Free Description of the FRT Construction 128
4.3 A(R)°p, A{R)c°p, and A{R)op cop as FRT Constructions 131
4.4 Conditions for A(R) to be a Pointed Bialgebra 138
5. THE FUNDAMENTAL EXAMPLE OF A QUANTUM GROUP 143
5.1 Review of SL(2,fc) 143
5.1.1 The Coordinate Ring of SL(2, A;) 144
5.1.2 The Lie Algebra sl(2,fc) 146
5.1.3 Irreducible Representations of sl(2, k) 148
5.2 Derivations and (Co)Algebra Actions Revisited 149
5.3 A Hopf Algebra Closely Related to fc[SL(2,fc)] 150
5.4 Grouplikes and Skew Primitives of fc[SL,(2, k)}° 151
5.5 Embedding W(sl(2, it)) into Ar[SL(2,fc)]° 153
5.6 Quantum Analogs of W(sl(2, A;)) 155
6. QUASITRIANGULAR STRUCTURES AND THE DOUBLE 161
6.1 Quasitriangular Algebras 161
6.2 Quasitriangular Structures Arising from Integrals 162
6.3 Quasitriangular Bialgebras and Quasitriangular Hopf Algebras 164
6.4 The Quantum Double 175
6.5 Some Fundamental Examples of Pointed Hopf Algebras 181
6.5.1 Q Binomial Coefficients 182
6.5.2 Construction of the Examples 184
6.6 A Family of QT Hopf Algebras and Associated Doubles 186
6.6.1 Construction and Properties of//(w.^lj) 187
6.6.2 Construction and Properties of f/fjv.^w) 191
7. COQUASITRIANGULAR STRUCTURES 197
7.1 Further Properties of A{R) 197
X INTRODUCTION TO THE QYBE
7.2 CoquasitriangularCoalgebras 199
7.3 Coquasitriangular Bialgebras and Hopf Algebras 203
7.4 The Free Coquasitriangular Bialgebra 209
7.5 One Parameter QYBE, Coquasitriangularity, and Tensor Product 213
7.5.1 /^ Commutative Spectral Parameter 214
7.5.2 Constructions when X is a Group 215
7.5.3 Tensor Product of One Parameter QYBE Solutions 218
8. SOME CLASSES OF SOLUTIONS 219
8.1 Some Consequences of M Reduction 220
8.2 When A(R) is Generated by Grouplike Elements 222
8.3 Solutions when DimM = 2 and A(R) is Pointed 226
8.4 Patching and Solutions in Higher Dimension 232
8.5 A Class of Weak QYB Modules 233
8.6 Some One Parameter Solutions 244
8.6.1 Some Specific Solutions 244
8.6.2 A p Perturbation Example 247
9. CATEGORICAL CONSTRUCTIONS 249
9.1 Coends 249
9.2 Quasi Symmetric Monoidal Categories 250
9.3 Rigid Monoidal Categories and Hopf Algebras 254
9.4 Categories and Coquasitriangular Hopf Algebras 258
9.5 The QYBE in Other Categories 258
9.6 The Category of Graded Modules 259
Appendices 261
A Prerequisites 261
A.1 The Ground Ring k and Basic fc Linear Maps 261
A.2 Algebras, Coalgebras, and Their Representations 262
A.3 Various Notations Related to the QYBE 263
A.3.1 Structure Constants 263
A.3.2 Heyneman Sweedler and H S Notations 267
A.3.3 Categorical Notation 268
A.4 Some Results from Linear Algebra 269
A.4.1 Rank of Tensors and Endomorphisms 269
A.4.2 Closed Subspaces of U* 272
A.4.3 Cofinite Subspaces and Continuous Linear Maps 277
References 281
Index 291
|
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| indexdate | 2024-07-09T18:26:21Z |
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| isbn | 0792347218 |
| language | English |
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| series2 | Mathematics and its applications |
| spelling | Lambe, Larry A. Verfasser aut Introduction to the quantum Yang Baxter equation and quantum groups an algebraic approach by Larry A. Lambe and David E. Radford Dordrecht [u.a.] Kluwer 1997 XX, 293 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 423 Mathematische Physik Hopf algebras Mathematical physics Quantum groups Yang-Baxter equation Quantengruppe (DE-588)4252437-4 gnd rswk-swf Yang-Baxter-Gleichung (DE-588)4291478-4 gnd rswk-swf Yang-Baxter-Gleichung (DE-588)4291478-4 s Quantengruppe (DE-588)4252437-4 s DE-604 Radford, David E. 1943- Verfasser (DE-588)1020225114 aut Mathematics and its applications 423 (DE-604)BV008163334 423 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008387756&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lambe, Larry A. Radford, David E. 1943- Introduction to the quantum Yang Baxter equation and quantum groups an algebraic approach Mathematics and its applications Mathematische Physik Hopf algebras Mathematical physics Quantum groups Yang-Baxter equation Quantengruppe (DE-588)4252437-4 gnd Yang-Baxter-Gleichung (DE-588)4291478-4 gnd |
| subject_GND | (DE-588)4252437-4 (DE-588)4291478-4 |
| title | Introduction to the quantum Yang Baxter equation and quantum groups an algebraic approach |
| title_auth | Introduction to the quantum Yang Baxter equation and quantum groups an algebraic approach |
| title_exact_search | Introduction to the quantum Yang Baxter equation and quantum groups an algebraic approach |
| title_full | Introduction to the quantum Yang Baxter equation and quantum groups an algebraic approach by Larry A. Lambe and David E. Radford |
| title_fullStr | Introduction to the quantum Yang Baxter equation and quantum groups an algebraic approach by Larry A. Lambe and David E. Radford |
| title_full_unstemmed | Introduction to the quantum Yang Baxter equation and quantum groups an algebraic approach by Larry A. Lambe and David E. Radford |
| title_short | Introduction to the quantum Yang Baxter equation and quantum groups |
| title_sort | introduction to the quantum yang baxter equation and quantum groups an algebraic approach |
| title_sub | an algebraic approach |
| topic | Mathematische Physik Hopf algebras Mathematical physics Quantum groups Yang-Baxter equation Quantengruppe (DE-588)4252437-4 gnd Yang-Baxter-Gleichung (DE-588)4291478-4 gnd |
| topic_facet | Mathematische Physik Hopf algebras Mathematical physics Quantum groups Yang-Baxter equation Quantengruppe Yang-Baxter-Gleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008387756&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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